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2017 is not just another prime number

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Re: 2017 is not just another prime number

#51

Post by ahakki »

Someone wrote: post #41
gus_massa wrote: post #34

Earlier quoted context omitted.

Technical note: 5 cubes is not enough for every number, so this property of 2017 is not trivial. From > Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes: > 23 = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3. I couldn't find if it's common tha…

http://oeis.org/A003328 : Numbers that are the sum of 5 positive cubes 5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 64, 68, 71, 75, 78, 82, 83, 89, 90, 94, 96, 97, 101, 108, 109, 115, 116, 120, 127, 129, 131, 134, 135, 136, 138, 143, 145, 146, 150, 152, 153, 155, 157, 162, 164, 169, 171, 172, 176, 181, 183, 188, 190, 192, 194 It seems this is fairly common (1757 is the 1000th such number), but of course that says n…

yes, but 2017 is the sum of five destinct integer cubes

Re: 2017 is not just another prime number

#53

Post by AccordionGuy »

williamstein wrote: post #17

Verifications of all the statements using SageMath, in case you want to be convinced or explore further: https://cloud.sagemath.com/projects/4a5f0542-5873-4eed-a85c-...

I tweaked the code to find the 5 integers whose cubes sum to 2017 to insure that they are distinct, and to pull out of all those loops once a soltuion has been found: https://cloud.sagemath.com/projects/d07701cd-2d97-4a09-96ef-...

Re: 2017 is not just another prime number

#54

Post by gweinberg »

aisofteng wrote: post #39
gcr wrote: post #6

Earlier quoted context omitted.

Every number is special. Proof is by contradiction: Assume that not every number is mathematically interesting and let X be the first such number. However, the fact that X is the lowest such number is itself pretty special, right?

This "proof" seems to be popular, and it always bothers me that it's invalid. It uses self-reference in an invalid way. Allow me to formalize; we take as a rigorous definition of an "interesting number" that a number has a unique property. Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. In formal first order logic, n is interesting if there exist a predicate P and a n…

I think it was always intended to be sort of a joke. "Interesting" is subjective, but obviously any self-consistent criteria of interesting cannot allow something to be interesting by it's sheer uninterstingness.

But even if you're willing to accept the paradox, it's still a bullshit proof. It's one thing to say a number is interesting because it really is the first number you can't think of anything interesting to say about it. It's something else to say to say it's interesting because its the first number you can't find anything interesting about, not counting all the others that were considered "interesting" for the same reason.

Re: 2017 is not just another prime number

#56

Post by s-phi-nl »

otalp wrote: post #21
warent wrote: post #20

Nice! Although I think it's amusing that they said "odd primes" as if there's any even primes

2 is a prime number...

   All throughout the talk there were statements like "Let p be an odd prime and…"
   
   My friend asked, “what is an odd prime?”—thinking it must be special in some way. The answer back was: not 2.
from https://rjlipton.wordpress.com/2009/05/18/boolean-solutions-...

Re: 2017 is not just another prime number

#57

Post by duomono4 »

* (loop for i from 1 to 15 do (loop for i2 from i to 15 do (loop for i3 from i2 to 15 do (loop for i4 from i3 to 15 do (loop for i5 from i4 to 15 do (if (= (+ (* i i i) (* i2 i2 i2) (* i3 i3 i3) (* i4 i4 i4) (* i5 i5 i5)) 2017) (print (list i i2 i3 i4 i5))))))))

(1 2 2 10 10) (1 2 4 6 12) (2 4 6 9 10) NIL

Re: 2017 is not just another prime number

#58

Post by mikeash »

wbhart wrote: post #12
mikeash wrote: post #8

Earlier quoted context omitted.

Can this proof be adapted for the reals, or is it only the case that every integer is special?

It's not a proof of anything, so it can't be adapted to the reals. Specialness has no rigorous mathematical definition, and certainly the second non-special number is not special, even if the first is. But more mathematically speaking, given a standard real number, there is no "next" real number. Induction is one of the defining properties of the natural numbers and it is induction that allows one to construct proofs…

I would argue that is is a proof of something, namely that you can't have a rigorous mathematical definition of specialness.

Re: 2017 is not just another prime number

#60

Post by gcr »

aisofteng wrote: post #39
gcr wrote: post #6

Earlier quoted context omitted.

Every number is special. Proof is by contradiction: Assume that not every number is mathematically interesting and let X be the first such number. However, the fact that X is the lowest such number is itself pretty special, right?

This "proof" seems to be popular, and it always bothers me that it's invalid. It uses self-reference in an invalid way. Allow me to formalize; we take as a rigorous definition of an "interesting number" that a number has a unique property. Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. In formal first order logic, n is interesting if there exist a predicate P and a n…

Thank you. As someone who struggles expressing concepts formally, it was enlightening to see how this sort of proof could be modeled.
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